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Photon sphere vs horizon as a null suface

  1. Sep 10, 2012 #1

    Jip

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    Hi ,
    There's something I don't get regarding the orbits of photons in Schwarzschild geometry.

    As well known, by solving geodesics equation for null rays, you get that photons can be in a (unstable) circular orbit at r=3 M. However, if you look at the causal diagram for Schwarschild geometry (in Schwarzschild coordinates, say), then you see that the light cones are still open.

    So, according to this diagram, a photon should go left (towards decreasing r) or right (increasing r), but not straight at r=cst!! Is the solution somewhere hidden in the two other coordinates theta and phi?

    I have another, but related question. The light cone gets degenerate with no extension at r=2M, as you see again on this causal diagram. In other words, the horizon is a null surface, which is a general property anyway. But if the horizon is a null surface, why don't photons actually move along this null surface? Instead, they just cross inwards.

    To summarize, I would have expected the photon sphere to be precisely the horizon, since for me I have this (wrong) idea in head : null surface <-> photon sphere

    Where is the mistake?

    Thanks a lot
     
  2. jcsd
  3. Sep 10, 2012 #2

    Bill_K

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    At the horizon r = 2M, the light rays (not "photons") that generate the null surface travel outward. Remaining at constant r = 2M of course.

    r = 3M is not a null surface. The light rays that go sideways travel in circular orbits.
     
    Last edited: Sep 10, 2012
  4. Sep 10, 2012 #3

    DrGreg

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    Yes. The diagram shows only radial motion (constant [itex]\theta[/itex] and [itex]\phi[/itex]) and so can't show orbits. The light cone is a 4-D cone and the diagram you refer to (I assume) shows only 2 dimensions.
     
  5. Sep 10, 2012 #4

    bcrowell

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    A light ray directed inward from the horizon travels inward. A light ray directed outward from the horizon maintains constant r=2M, so on a Penrose diagram it just travels along that diagonal line until it gets to [itex]\mathscr{I}^+[/itex].
     
  6. Sep 11, 2012 #5

    Jip

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    Hi,
    Thanks for the replies. Now I understand better.

    -For the photon sphere, a light ray can travel inward or outward at r=3M, but it might also somehow balance gravity with some radial speed + an orthoradial speed (roughly speaking, it's just a way to visualize what's going on; I think the anology with an accoustic black hole helps here, where I guess you can have a circular sonic wave orbiting around the sonic horizon (?))

    -Regarding the light cones at r=2M. I think I was fooled by the use of Schwarzschild coordinates. In these coordinates, the light cone schrinks and has no extension anymore at r=2M. So, naively, you think that light rays must stay at r=2M. However in advanced Eddigthon-Finkelstein coordinates, you rather see that the light cone is not degenerate, and that, as you say, the light ray outward stays at r=2M, all the others fall towards the singularity. I have to see this as well on the Penrose diagram.

    My confusion came from the fact that if you stay at r=2M, then you are in a circular orbit. But it is wrong. You can be static as well!

    So, if I understood correctly, there are null geodesics that stay forever at some point $theta_0$, $\phi_0$, $r=2M$, $\forall t$. Is that right?
     
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